Lógica AULA 9_respostas

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AULA 09 - RESPOSTAS 
 
Verificar que são válidos os seguintes argumentos: 
 
1) r → p v q, r, ~p| q 
 
1. r → p ˅ q 
2. r 
3. ~ p . 
4. p ˅ q 1,2: MP 
5. q 3,4: SD 
 
 
2) p → ~ q, ~~ q, ~p → r | r 
 
1. p → ~ q 
2. ~ ~ q 
3. ~ p → r . 
4. ~ p 2,3: MT 
5. r 3,4: MP 
 
3) p ∧ q, p → r, q → s | r ∧ s 
 
1. p ∧ q 
2. p → r 
3. q → s . 
4. p 1: Simp. 
5. q 1: Simp. 
6. r 2,4: MP 
7. s 3,5: MP 
8. r ∧ s 6,7: Conjunção 
 
4) p→ q, q → ~r, p | ~r 
 
1. p→ q 
2. q → ~r 
3. p . 
4. q 1,3: MP 
5. ~r 4,2: MP 
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5) p → q, ~q, ~p →r | r 
 
 
1. p → q 
2. ~q 
3. ~p →r 
4. ~p 1,2: MT 
5. r 4,3: MP 
 
6) p → q, p → r, p | q ∧ r 
 
1. p → q 
2. p → r 
3. p . 
4. r 2,3: MP 
5. q 1,3: MP 
6. q ∧ r 4,5: Conjunção 
 
7) p → q, ~q, p v r | r 
 
1. p → q 
2. ~q 
3. p ˅ r . 
4. ~p 1,2: MT 
5. r 3,4: SD 
 
8) p v ~q, r → ~p, r | ~q 
1. p ˅ ~q 
2. r → ~p 
3. r . 
4. ~p 2,3: MP 
5. ~q 1,4: SD 
 
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9) ~p v ~q, ~~ q, r → p | ~r 
1. ~p ˅ ~q 
2. ~ ~ q 
3. r → p . 
4. ~ p 1,2: SD 
5. ~ r 3,4: MT 
 
10) p → ~q, ~~ q, ~p →r v s | r v s 
 
1. p → ~q 
2. ~ ~ q 
3. ~ p → r ˅ s . 
4. ~ p 1,2: MT 
5. r ˅ s 3,4: MP 
 
11) ~p ˅ ~ ~ q, ~ ~ p, ~ r → ~q | ~ ~ r 
 
1. ~ p ˅ ~ ~ q 
2. ~ ~ p 
3. ~ r → ~ q . 
4. ~ ~ q 1,2: SD 
5. ~ ~ r 3,4: MT 
 
12) p → ~ q ∧ r, p, s → q, s ˅ t | t 
1. p → ~ ∧ r 
2. p 
3. s → q 
4. s ˅ t . 
5. ~ q ∧ r 1,2: MP 
6. ~ q 5: Simp. 
7. ~ s 3,6: MT 
8. t 4,7: SD 
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13) p ∧ q, p → r, r ∧ s → ~ t, q → s | ~ t 
 
1. p ∧ q 
2. p → r 
3. r ∧ s → ~t 
4. q → s . 
5. p 1: Simp. 
6. q 1: Simp. 
7. r 2,5: MP 
8. s 4,6: MP 
9. r ∧ s 7,8: SD 
10. ~ t 9,3: MP 
 
14) p ∧ ~q, q v ~r, s → r | p ∧ ~s 
 
1. p ∧ ~q 
2. q ˅ ~r 
3. s → r . 
4. ~ q 1: Simp. 
5. p 1: Simp. 
6. ~ r 2,4: SD 
7. ~ s 3,6: MT 
8. p ∧ ~ s 5,7: Conjunção 
 
15) p ˅ ~ q, ~ q → r, p → s, ~ r | s 
 
1. p ˅ ~ q 
2. ~ q → r 
3. p → s 
4. ~ r . 
5. q 2,4: MT 
6. p 1,5: SD 
7. s 3,6: MP 
 
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16) p → q, q → ~ r, ~ ~ r, p ˅ (s ∧ t) | s 
 
1. p → q 
2. q → ~ r 
3. ~ ~ r 
4. p ˅ (s ∧ t) . 
5. ~ q 2,3: MT 
6. ~ p 1,5: MT 
7. s ∧ t 4,6: SD 
8. s 7: Simp. 
 
17) p ˅ q, q → r, p → s, ~ s | r ∧ (p ˅ q) 
 
1. p ˅ q 
2. q → r 
3. p → s 
4. ~ s . 
5. r ˅ s 1,2,3: DC 
6. r 4,5: SD 
7. r ∧ (p ˅ q) 1,6: Conjunção 
 
18) ~ p ˅ ~ q, ~ q → ~ r, ~ p → t, ~ t | ~ r ∧ ~ t 
 
1. ~ p ˅ ~ q 
2. ~ q → ~ r 
3. ~ p → t 
4. ~ t . 
5. ~ r v t 1,2,3: DC 
6. ~ r 4,5: SD 
7. ~ r ∧ ~ t 4,6: Conjunção 
 
 
 
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19) p → ~ q, p ˅ r, r → ~ q, s → q, t | ~ s ∧ t 
 
1. p → ~q 
2. p ˅ r 
3. r → ~ q 
4. s → q 
5. t . 
6. ~ q ˅ ~ q 1,2,3: DC 
7. ~ q 6: Idempotência 
8. ~ s 4,7: MT 
9. ~ s ∧ t 8,5: Conjunção 
 
20) ~ p → q, q → r ∧ s, p → t, ~ t | s 
 
1. ~ p → q 
2. q → r ∧ s 
3. p → t 
4. ~ t . 
5. ~ p 3,4: MT 
6. q 1,5: MP 
7. r ∧ s 2,6: MP 
8. s 7: Simp. 
 
21) p v q, q → r, ~r | p 
 
1. p ˅ q 
2. q → r 
3. ~ r . 
4. ~ q 2,3: MT 
5. p 1,4: SD 
 
 
 
 
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22) p → q, ~ q ∧ ~ r, ~ r → s | ~ p ∧ s 
 
1. p → q 
2. ~ q ∧ ~ r 
3. ~ r → s . 
4. ~ q 2: Simp. 
5. ~ r 2: Simp. 
6. ~ p 1,4: MT 
7. s 3,5: MP 
8. ~ p ∧ s 6,7: Conjunção 
 
 
 
23) p → q, p v r, ~r | q v s 
 
1. p → q 
2. p ˅ r 
3. ~ r . 
4. p 2,3: SD 
5. q 1,4: MP 
6. q ˅ s 5: Adição 
 
 
 
24) p → q, q → r, (p →r)→ ~ s, s ˅ t | t 
 
1. p → q 
2. q → r 
3. (p →r)→ ~s 
4. s ˅ t . 
5. p → r 1,2: SH 
6. ~s 3,5: MP 
7. t 4,6: SD 
 
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25) p ˅ ~q, ~r, p → r, ~q → s | s 
 
1. p ˅ ~q 
2. ~r 
3. p → r 
4. ~q → s . 
5. r ˅ s 1,3,4: DC 
6. s 2,5: SD 
 
 
26) r → t, s → q, t ˅ q → ~p, r ˅ s | ~ p 
 
1. r → t 
2. s → q 
3. t ˅ q → ~p 
4. r ˅ s . 
5. t ˅ q 1,2,4: DC 
6. ~ p 3,5: MP 
 
 
27) p → ~ q, ~ q → ~ s, (p → ~ s) → ~ t, r → t | ~ r 
 
1. p → ~ q 
2. ~ q → ~ s 
3. (p → ~s) → ~t 
4. r → t . 
5. p → ~ s 1,2: SH 
6. ~ t 3,5: MP 
7. ~ r 4,6: MT 
 
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28) p ˅ q → ~ r, s → p, t → q, s v t | u ˅~r 
1. p ˅ q → ~r 
2. s → p 
3. t → q 
4. s ˅ t . 
5. p ˅ q 2,3,4: DC 
6. ~ r 1,5: MP 
7. u ˅ ~ r 6: Adição 
 
29) p ˅ q → ~ r, p, s→ r | ~ s 
1. p ˅ q → ~ r 
2. p 
3. s→ r . 
4. p ˅ q 2: Adição 
5. ~ r 1,4: MP 
6. ~ s 3,5: MT 
 
30) p ∧ (q ˅ r), q ˅ r → ~ s, s ˅ t | t 
 
1. p ∧ (q ˅ r) 
2. q ˅ r → ~s 
3. s ˅ t . 
4. q ˅ r 1: Simp. 
5. ~ s 2,4: MP 
6. t 3,5: SD 
 
31) p ˅ q → ~r, q, s ∧ t → r | ~ (s ∧ t) 
 
1. p ˅ q → ~r 
2. q 
3. s ∧ t → r . 
4. p ˅ q 2: Adição 
5. ~r 1,4: MP 
6. ~(s ∧ t) 3,5: MT 
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32) p → q, ~q, ~p v ~r → s | s 
 
1. p → q 
2. ~ q 
3. ~ p ˅ ~ r → s 
4. ~ p 1,2: MT 
5. ~ p ˅ ~ r 4: Adição 
6. s 3,5: MP 
 
33) p ˅ (q ∧ r), q → s, r → t, s ∧ t → p˅ r, ~ p | r 
 
1. p ˅ (q ∧ r) 
2. q → s 
3. r → t 
4. s ∧ t → p ˅ r 
5. ~p . 
6. q ∧ r 1,5: SD 
7. q 6: Simp. 
8. r 6: Simp. 
9. s 2,7: MP 
10. t 3,9: MP 
11. s ∧ t 9,10: Conjunção 
12. p ˅ r 11,4: MP 
13. r 12,5: SD 
 
34) q ˅ (r → t), q → s, ~ s → (t → p), ~ s | r → p 
1. q ˅ (r → t) 
2. q → s 
3. ~ s → (t → p) 
4.~ s . 
5. t → p 3,4: MP 
6. ~ q 2,4: MT 
7. r → t 1,6: SD 
8. r → p 5,7: SH 
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35) p ˅ q → (p → s ∧ t), p ∧ r | t ˅ u 
 
1. p ˅ q → (p → s ∧ t) 
2. p ∧ r . 
3. p 2: Simp. 
4. p ˅ q 3: Adição 
5. p → s ∧ t 1,4: MP 
6. s ∧ t 3,5: MP 
7. t 6: Simp. 
8. t ˅ u 7: Adição